I think that it's Will himself rather than the firm - I think that there have been foreshadowings that Will has dodgy dealings in the past. Kalinda's antics are more out in the open though, so an easy route into LG for Childs.
Peter wants to keep the guy at the DA's office onside as he knows about Peter and Kalinda. He removed pages from investigation notes so that Childs wouldn't find out about it (hedging his bets to keep a job when Peter gets in). He knows that Peter won't sack him whilst he has that knowledge.
I think that the Blake/Kalinda/Peter thing is very murky. Kalinda used to be someone else (though it is hard with the wibbling of AP's accent to tell whether she is supposed to be originally British or US/Canadian, so we can't know how far she has run). She was this person when she started at the DA's office, and then at some point changed her name.
It has all been messed up though by the badly thought out Kalinda/Blake story, which the writers thought would be this great UST thing, but has just been awful and fucked up.
My speculation about the Kalinda thing is this:
Kalinda was married (possibly arranged marriage). Either something happened to her (she lost a baby, she accidentally killed someone, her husband abused her), or possibly her family found out about her wavering sexuality and disowned her. She runs for it, wants to change her name. Along comes Peter - and he agrees to help her change her name and make her old identity go away, for a price (sex - because he is such a keeper).